Every person who works in the United States must pay taxes on their earnings (regardless of their status). That said, it does not sound like you are making enough money so that you will owe the government anything. Regardless of that fact it is always best to file your taxes every single year. You can do this late if needed and in your case I would recommend that you go and file now for the missing years.
Whether or not you pay taxes is something the government will look at when making decisions on green card and citizenship applications. Because of this you should file taxes every year going forward.
You should contact an accountant or other person that prepares income tax filings. You should file your income tax return whether or not you have income that meets the threshold or paying taxes. You should also contact an immigration attorney for filing the permanent resident application.
Just because you have been granted asylum does not make you exempt from paying taxes. Consult with an accountant to determine your tax liability and filing requirements with the IRS.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Call +1-561-478-5353 to schedule a consultation with Mr. Devore.